RDP 2018-08: Econometric Perspectives on Economic Measurement Appendix A: The Inconsistency Arising from Endogenous Weighting is a Linear Projection

Let δ and x_tv be vector shorthand for the full set of coefficients and regressors that are implicit in The Standard Model, for a given (t, v) pair. Then

Applying the Law of Large Numbers and the Continuous Mapping Theorem (see Hansen (2018)),

The right-hand side of Equation (A3) is a weighted linear projection of the errors on the regressors. Since the weights are functions of errors, the second expectation term is not zero.